- A function
*f*is an**even function**if:- its graph is symmetric with respect to the vertical axis,
meaning that (−
*a*,*b*) is on the graph whenever (*a*,*b*) is; *f*(−*x*) =*f*(*x*) whenever*f*(*x*) exists.

- its graph is symmetric with respect to the vertical axis,
meaning that (−
- A function
*f*is an**odd function**if:- its graph is symmetric with respect to the origin,
meaning that (−
*a*, −*b*) is on the graph whenever (*a*,*b*) is; *f*(−*x*) = −*f*(*x*) whenever*f*(*x*) exists.

- its graph is symmetric with respect to the origin,
meaning that (−
- A function
*f*is a**zero function**if:- its graph is symmetric with respect to the horizontal axis,
meaning that (
*a*, −*b*) is on the graph whenever (*a*,*b*) is; *f*(*x*) = 0 whenever*f*(*x*) exists.

- its graph is symmetric with respect to the horizontal axis,
meaning that (

*f*(*x*) = 0.

The graph of a function can only have one vertical intercept, which is easy to compute:

- (0,
*f*(0)).

*f*(*x*) =*m**x*+*b*.

Few functions are linear.
However, given two inputs *a* and *b* of the function
(and assuming that the function is defined between *a* and *b*),
we can imagine a line drawn through the corresponding points on the graph,
which is called a **secant** line,
(so called from the Latin for ‘cutting’)
and calculate its slope:

*m*= [*f*(*b*) −*f*(*a*)] ÷ [*b*−*a*].

For a particular function on a particular nontrivial interval, sometimes the average rate of change of that function between any two points in that interval always has the same sign (positive or negative).

- The function is
**increasing**on the interval if the average rate of change is always positive. - The function is
**decreasing**on the interval if the average rate of change is always negative. - The function is
**constant**on the interval if the average rate of change is always zero.

A function has a **local maximum** at an input *b*
if it is increasing on an interval [*a*, *b*]
and decreasing on an interval [*b*, *c*].
It has a **local minimum** at *b*
if it is decreasing on an interval [*a*, *b*]
and increasing on an interval [*b*, *c*].
A **local extremum** (also called a *relative* extremum)
is anything that's either a local maximum or a local minimum.
(To be very precise, these are really *strict two-sided* local extrema.
If you allow the functions to be weakly increasing or decreasing to the sides,
then you get *weak* local extrema;
if you allow them to be undefined on one side,
then you get *one-sided* local extrema.)

A function has an **absolute maximum** at an input *b*
if *f*(*a*) ≤ *f*(*b*)
for any *a* in the domain of *f*.
It has an **absolute minimum** at *b*
if *f*(*a*) ≥ *f*(*b*)
for any *a* in the domain of *f*.
An **absolute extremum** (also called a *global* extremum)
is anything that's either an absolute maximum or an absolute minimum.
(This is the weakest and default meaning of these terms,
although sometimes people talk about strict or two-sided absolute extrema.)

Note the grammar used with this terminology (both local and absolute):
when a function *f* has an extremum *at* *c*,
the extremum *is* *f*(*c*).
Calculating extrema is much easier with Calculus,
so mostly you'll just look for them on the graph in this course.
(The traditional plurals of all of these ‘‑um’ words
end in ‘‑a’;
again, this comes from Latin.)

Go back to the course homepage.

This web page was written by Toby Bartels, last edited on 2018 August 1. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2020SP/fnprops/`

.